Quasiperiodic waves and asymptotic behavior for Bogoyavlenskii's breaking soliton equation in (2+1) dimensions.

Based on a multidimensional Riemann theta function, the Hirota bilinear method is extended to explicitly construct multiperiodic (quasiperiodic) wave solutions for the (2+1) -dimensional Bogoyavlenskii breaking soliton equation. Among these periodic waves, the one-periodic waves are well-known cnoidal waves, their surface pattern is one-dimensional, and often they are used as one-dimensional models of periodic waves in shallow water. The two-periodic (biperiodic) waves are a direct generalization of one-periodic waves, their surface pattern is two dimensional, that is, they have two independent spatial periods in two independent horizontal directions. The two-periodic waves may be considered to represent periodic waves in shallow water without the assumption of one dimensionality. A limiting procedure is presented to analyze asymptotic behavior of the one- and two-periodic waves in details. The exact relations between the periodic wave solutions and the well-known soliton solutions are established. It is rigorously shown that the periodic wave solutions tend to the soliton solutions under a "small amplitude" limit.